A point (4,-2) lies on the terminal arm of an angle in standard position.
a) What is the value of the principal angle (theta) to the nearest degree? Related acute angle to the nearest degree?
b) state the six trigonometric ratios as exact values.
a) To find the principal angle, we need to find the angle whose terminal arm contains the point (4, -2).
The principal angle is the angle formed between the positive x-axis and the line connecting the origin to the point (4, -2). We can find the principal angle using the arctan function.
The principal angle (theta) = arctan(-2/4)
= arctan(-1/2)
Using a scientific calculator, we find that the principal angle is approximately -26.57 degrees.
To find the related acute angle, we take the absolute value of the principal angle:
Related acute angle = |-26.57|
= 26.57 degrees (to the nearest degree).
b) The six trigonometric ratios are as follows:
sin(theta) = opposite/hypotenuse = -2/√(4^2 + (-2)^2) = -2/√20 = -√5/5
cos(theta) = adjacent/hypotenuse = 4/√(4^2 + (-2)^2) = 4/√20 = 2√5/5
tan(theta) = opposite/adjacent = -2/4 = -1/2
csc(theta) = 1/sin(theta) = -5/√5 = -√5
sec(theta) = 1/cos(theta) = √5/2√5 = 1/2
cot(theta) = 1/tan(theta) = -2/1 = -2